One can clearly see that these six types can be grouped into two categories.
The first category is that one component is completely separated from the other two. Though this may rarely happen for real data, the benefit brought by this property is significant because it allows us to write the covariance matrix as a Kronecker product of two smaller matrices.
The three cases in this category are V\(\mid \) ST, S\(\mid \) VT, and T\(\mid \) VS.
For example, when the covariance is V\(\mid \) ST, it can be decomposed into two parts corresponding to the correlation among variables (\(C_{ij}(\mathbf {0},0)\) ) and the spatio-temporal correlation (\(\rho _1(\mathbf {h},u)\) ). This is also known as the intrinsic correlation model in [1]}. We know \(C_{ij}(\mathbf {h},u) = \rho _1(\mathbf {h},u)C_{ij}(\mathbf {0},0) = \rho _1(\mathbf {h},u)C_{ji}(\mathbf {0},0) = C_{ji}(\mathbf {h},u)\) , so the multivariate spatio-temporal covariance is also \(\text{V}^\text{sym}\) .
